Local methods for receptivity prediction of stationary crossflow mode to micron-sized distributed roughness

Two locally solvable methods are developed for receptivity prediction of stationary crossflow mode to micron-sized distributed roughness. The first approach is established based on Linear Stability Theory (LST) and its adjoint equations (ALST). Distributed receptivity can be treated as a superposition of local receptivity solutions, while significant cumulative errors are also introduced due to the neglected boundary layer non-parallelism. To address this limitation, an alternative methodology is proposed by employing the Expansion of the Parabolized Stability Equations (EPSE) and their adjoint equations (AEPSE). Derived from Linear Parabolized Stability Equations (LPSE) and the adjoint counterpart (ALPSE) via first-order Taylor expansion, the EPSE and AEPSE enable localized solutions. Validation across three quasi three-dimensional (3D) boundary layer cases demonstrates that AEPSE method achieves significantly better agreement with ALPSE and Adjoint Harmonic Linear Navier-Stokes equations (AHLNS) benchmarks than ALST method. The development of local receptivity analysis methods is motivated by integrating them with established N-factor methods, such as the saddle point method or ray tracing technique. Such integration is expected to facilitate amplitude-based transition prediction methods for general 3D boundary layers on practical aircraft surfaces. Although the computational robustness of current EPSE implementations is found to be inferior to that of LST, it is contended that with future algorithmic optimizations, significant potential for engineering applications can be retained by the AEPSE method.